BuraliFortiParadox 2002-09-28 19:13:57 2006-08-16 18:03:57 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} The \emph{Burali-Forti} paradox demonstrates that the class of all ordinals is not a set. If there were a set of all ordinals, $Ord$, then it would follow that $Ord$ was itself an ordinal, and therefore that $Ord\in Ord$. \PMlinkescapetext{Even} if sets in general are allowed to contain themselves, ordinals cannot since they are defined so that $\in$ is well founded over them. This paradox is similar to both Russell's paradox and Cantor's paradox, although it predates both. All of these paradoxes prove that a certain object is ``too large'' to be a set.